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Re: Vol of cylinder intersected by 2 planes
Posted:
Nov 30, 2002 11:34 PM


Calculus can be used to solve this problem:
We have along the zaxis the movement from Z1 to Z'. We take infinitesimal slices parallel to the xyplane
Along the xaxis we have the movement from some value of x depending upon ratio of Z'/Z1Z0 to x = D on the right. I think the proper function for x on the left is x = D * (2Z'Z1Z0)/(Z1Z0). You can see that if Z' = Z0, then x = D, which is expected for a full circle. If Z' = Z1+Z0/2 or halfway across, then x = 0, a semicircle as expected and if Z' = Z1, then x = D or at the right hand side. It is important to realize that we're taking the area of circle segments, and the circle is segmented by a vertical line that depends upon the value of Z' at the moment. We consider the area of the circle to the right of this vertical line in the xyplane
However this value of starting x will decrease as we move along the z axis from Z' to Z1, where x = D and the area will decrease to 0.
Along the yaxis we have the movement from y to +y where y is either the bottom or top of a circle of radius D. We can simplify y to run from 0 to +y at the top of the circle and double this area.
If I am not making a mistake, I believe the integral is
for z = Z' to Z1 the integral of ( for x = D*(2zZ1Z0)/(Z1Z0) to D ( 2 * integral of (for y = 0 to sqrt(D^2x^2) dy ) ) dx ) dz
Can someone who has a symbolic algebra program check this? I hope the triple integral is readable.



